Very little theoretical work has focussed on HIV-1 spread in tissue culture. We used two systems of ordinary differential equations to model two modes of viral spread, cell-free virus and cell-to-cell contact. The models produce remarkably similar results. Simulations using realistic parameter regimes showed that starting with a small fraction of cells infected both cell- free viral spread and direct cell-to-cell transmission give an initial exponential phase of viral growth, followed by either a crash or a gradual decline extinguishing the culture. Under some conditions, surprisingly, an oscillatory phase may precede the extinction. After modeling, we later discovered experimental data showing that tissue infections can display several sequential cycles of oscillation. Significantly, therefore, infective oscillations can be explained by infection dynamics, biological explanations may be unnecessary. Quantitative fitting of a theory of this detail to experiment appears impractical, however. Thus, to pursue practical comparison of models and experiment, a theory describing a single cycle of viral replication will be developed. Suitable experimental techniques for determining the parameters in the theory will then be identified.